Lec 20: Path independence and conservative fields | MIT 18.02 Multivariable Calculus, Fall 2007

The script begins with an introduction to line integrals, specifically focusing on the concept of work done by a vector field along a curve. It explains the process of computing line integrals in the context of work, using the integral of a vector field F dot dr, or in terms of a unit tangent vector and arc length element. An example is introduced where the vector field yi + xj is considered, and the work done along a closed curve starting and ending at the origin is to be computed. The curve consists of three parts, enclosing a sector of a unit disk, and the computation involves setting up integrals for each part of the curve.

This paragraph delves into the geometrical and parameterical computation of line integrals. It starts with a discussion of the vector field's direction on the x-axis and how it is perpendicular to the curve, leading to zero work done. The script then moves on to the second part of the curve, a portion of the unit circle, and explains how to parameterize x and y in terms of the angle theta. The integral is computed by substituting dx and dy with their respective derivatives in terms of theta, resulting in the integral of cosine squared minus sine squared, which simplifies to cosine of two theta. The computation concludes with the integral from zero to pi/4, yielding a result of one half.

The script continues with the third part of the curve, which is a diagonal line segment from (1/root two, 1/root two) back to the origin. It discusses the complexity of parameterizing this segment and suggests a simpler approach by considering the work done in one direction and then inferring the work in the opposite direction. The parameterization x = t, y = t is introduced for t from zero to one over root two, and the integral of 2t dt is computed, resulting in one half. The script then explains how to adjust this result for the actual path C3, either by reversing the parameterization or by taking the negative of the integral, leading to a total work of negative one half for the path C3.

The script introduces a special case of line integrals where the vector field is the gradient of a potential function, leading to simplifications in the computation of work. It explains that if the vector field is a gradient field, the line integral of the field along a curve from point P0 to P1 is simply the value of the potential function at P1 minus the value at P0. This is a direct application of the fundamental theorem of calculus to line integrals, which states that integrating a derivative recovers the original function. The script emphasizes that this simplification only applies to gradient fields and not to arbitrary vector fields.

This paragraph explores the concept of gradient fields and potential functions further. It discusses the physical interpretation of potential functions, such as electrical or gravitational potential, and the relationship between the gradient vector and the force field. The script clarifies the convention used in mathematics where the force field is the negative of the gradient of the potential function, which is a point of confusion between mathematicians and physicists. It also touches on the idea that the work done by the force field is equivalent to the change in potential energy from one point to another.

The script provides a proof for the fundamental theorem of calculus for line integrals in the context of gradient fields. It explains that by parameterizing a curve and expressing the differentials dx and dy in terms of a parameter t, the integral of the gradient field components f_sub_x dx + f_sub_y dy can be seen as the integral of df/dt, which is the rate of change of the potential function f with respect to t. The proof concludes by applying the fundamental theorem of calculus to show that the line integral is equal to the change in the potential function from the initial to the final point on the curve.

This paragraph discusses the property of path independence for gradient fields. It states that if a vector field is a gradient field, the work done along any path between two points is the same, regardless of the path taken. The script provides a proof for this property by showing that the line integral along a closed curve (where the start and end points are the same) is zero, which implies that the work done along any path can be inferred from the initial and final points alone, without regard to the path's shape or direction.

The script contrasts gradient fields with non-gradient fields, using a specific example of a vector field that rotates around the origin. It explains that for non-gradient fields, the work done along a closed curve is not necessarily zero, and thus they are not conservative. The example given is a vector field that, when integrated over a unit circle, results in a non-zero value, demonstrating that it is not conservative and cannot be the gradient of a potential function.

This paragraph connects the mathematical concept of conservative fields to their physical implications, particularly in the context of energy conservation. It explains that conservative fields, such as gravitational or electric fields, cannot provide perpetual motion or free energy, as the work done along any closed path in such a field is zero. The script also humorously warns against the dangers of electrical potential, commonly known as voltage.

The script concludes by summarizing the equivalent properties of conservative, path independent, and gradient fields. It states that a field being conservative (work done along any closed curve is zero), path independent (work done along any path between two points is the same), and being a gradient field are all equivalent properties. The paragraph also introduces the concept of an exact differential, which is a condition that is met when a differential can be expressed as the differential of a scalar function, indicating that the field is a gradient field.